where the second equality follows from the requirement (a) that \(\mathbf {\Omega }\)  be closed. This condition implies that the one-form \(\tilde{\mathbf {V}} = \mathbf {\Omega }\left(\bar{\mathbf {V}}\right)\)  is also closed, i.e. \(\tilde{\mathbf {d}}\tilde{\mathbf {V}} = 0\) . Clearly this is satisfied if \(\tilde{\mathbf {V}}\)  is exact, that is, \(\tilde{\mathbf {V}} = \tilde{\mathbf {d}}\tilde{G}[X]\) , for some e-functional \(\tilde{G}[X]\)  since \(\tilde{\mathbf {d}}^{2}\tilde{G}=0\)  identically. As is known, however, a closed form need not be exact; this is always true locally, but the global statement depends on the topology of \(\Gamma \) . Here we go further and assume that the topology of \(\Gamma \)  is such that it allows the identification of \(\tilde{\mathbf {V}}=\tilde{\mathbf {d}}\tilde{G}[X]\)  always.This is consistent, for example, with the fact that the space of pure states in quantum theory is a simply connected Kähler manifold. See e.g., [1]}. As one might expect, the e-functionals \(\tilde{G}\)  are special, they are the Hamiltonian generators.
